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Help: Sequence and Series 2

The sixth term of an arithmetic progression is 2, and its common difference is greater than 1. Show that the value of the common difference of the progression so that the product of the first, fourth and fifth therm is greatest is \(\frac{8}{5}\).

I did it with the concept of Maxima and Minima. Can someone suggest any other methods, something that is purely Algebra

Comments

Try to use weighted AM-GM, but keep in mind that \( a_1, a_4 \) are both negative, so their weights must be negative as well.
Take cases in which \( a_5 \) is positive or negative, and use the fact that \( a_6 = 2 \) for the weighted AM part.
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Ameya Daigavane
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1 year, 3 months ago

As am on mobile right now so couldn't post another post for this one so would you please consider it and clear my doubt here only.

A straight line is drawn through the centre of a square \(ABCD\) intersecting side \(AB\) at \(N\) so that \(AN:NB=1:2\). On this line take an arbitrary point \(M\) lying inside the square. Prove that the distances from the point \(M\) to the sides \(AB, AD, BC, CD\) of the square taken in that order, form an \(A.P.\)

I did it by considering any vertex as origin then considering sides common to it as axes and then calculating the distance of the point from the sides and then find out the common difference.

I would appreciate it if someone would tell me how to do a rigorous proof which would teach my some interesting new things given that I am a noob at coordinate geometry.

@Deeparaj Bhat
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Can you please look into the question in the original post, and provide me some other way to solve than the one that I did as stated in the note, or you could help me figure out where I am going wrong in the method suggested by @Ameya Daigavane.
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Akhilesh Prasad
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1 year, 3 months ago

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@Ameya Daigavane
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I did it the exact same way, just with one little deviation which was I selected the length of the sides to be \(a\) instead of \(3a\), it made things just a little harder to visualize. And did you think up anything about the question in the main post, or could you suggest me some text to read it from.
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Akhilesh Prasad
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1 year, 3 months ago